In practice, even storing the matrices that show up in the problem formulation can be prohibitive, let alone running a solver. Often you need to take advantage of sparsity in a nontrivial way (see e.g. COSMO [1]), or take advantage of cases where the solution can be approximated by a low rank matrix, and recently there has been a trend in using first-order methods like ADMM rather than full interior-point algorithms. The review article [2] from 2019 gives a feel for how much work has gone into making large-scale semidefinite programming possible, but has this sentence in their conclusion section:

"Semidefinite programming is still far from being a mature technology like linear or quadratic programming."

[1] https://link.springer.com/article/10.1007/s10957-021-01896-x [2] https://www.annualreviews.org/doi/pdf/10.1146/annurev-contro...